Astronomy:Cepheid variable

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Short description: Type of variable star that pulsates radially
RS Puppis, one of the brightest known Cepheid variable stars in the Milky Way galaxy
(Hubble Space Telescope)

A Cepheid variable (/ˈsɛfi.ɪd, ˈsiːfi-/) is a type of variable star that pulsates radially, varying in both diameter and temperature. It changes in brightness, with a well-defined stable period and amplitude.

Cepheids are important cosmic benchmarks for scaling galactic and extragalactic distances. A strong direct relationship exists between a Cepheid variable's luminosity and its pulsation period.

This characteristic of classical Cepheids was discovered in 1908 by Henrietta Swan Leavitt after studying thousands of variable stars in the Magellanic Clouds. The discovery allows one to know the true luminosity of a Cepheid by just observing its pulsation period. This tells one the distance to the star, by comparing its known luminosity to its observed brightness.

The term Cepheid originates from Delta Cephei in the constellation Cepheus, identified by John Goodricke in 1784. It was the first of its type to be identified.

History

The period-luminosity curves of classic and type II Cepheids

On September 10, 1784, Edward Pigott detected the variability of Eta Aquilae, the first known representative of the class of classical Cepheid variables.[1] The eponymous star for classical Cepheids, Delta Cephei, was discovered to be variable by John Goodricke a few months later.[2] The number of similar variables grew to several dozen by the end of the 19th century, and they were referred to as a class as Cepheids.[3] Most of the Cepheids were known from the distinctive light curve shapes with the rapid increase in brightness and a hump, but some with more symmetrical light curves were known as Geminids after the prototype ζ Geminorum.[4]

A relationship between the period and luminosity for classical Cepheids was discovered in 1908 by Henrietta Swan Leavitt in an investigation of thousands of variable stars in the Magellanic Clouds.[5] She published it in 1912 with further evidence.[6] Cepheid variables were found to show radial velocity variation with the same period as the luminosity variation, and initially this was interpreted as evidence that these stars were part of a binary system. However, in 1914, Harlow Shapley demonstrated that this idea should be abandoned.[7] Two years later, Shapley and others had discovered that Cepheid variables changed their spectral types over the course of a cycle.[8]

In 1913, Ejnar Hertzsprung attempted to find distances to 13 Cepheids using their motion through the sky.[9] (His results would later require revision.) In 1918, Harlow Shapley used Cepheids to place initial constraints on the size and shape of the Milky Way and of the placement of the Sun within it.[10] In 1924, Edwin Hubble established the distance to classical Cepheid variables in the Andromeda Galaxy, until then known as the "Andromeda Nebula" and showed that those variables were not members of the Milky Way. Hubble's finding settled the question raised in the "Great Debate" of whether the Milky Way represented the entire Universe or was merely one of many galaxies in the Universe.[11]

In 1929, Hubble and Milton L. Humason formulated what is now known as Hubble's Law by combining Cepheid distances to several galaxies with Vesto Slipher's measurements of the speed at which those galaxies recede from us. They discovered that the Universe is expanding, confirming the theories of Georges Lemaître.[12]

Illustration of Cepheid variables (red dots) at the center of the Milky Way[13]

In the mid 20th century, significant problems with the astronomical distance scale were resolved by dividing the Cepheids into different classes with very different properties. In the 1940s, Walter Baade recognized two separate populations of Cepheids (classical and type II). Classical Cepheids are younger and more massive population I stars, whereas type II Cepheids are older, fainter Population II stars.[14] Classical Cepheids and type II Cepheids follow different period-luminosity relationships. The luminosity of type II Cepheids is, on average, less than classical Cepheids by about 1.5 magnitudes (but still brighter than RR Lyrae stars). Baade's seminal discovery led to a twofold increase in the distance to M31, and the extragalactic distance scale.[15][16] RR Lyrae stars, then known as Cluster Variables, were recognized fairly early as being a separate class of variable, due in part to their short periods.[17][18]

The mechanics of stellar pulsation as a heat-engine was proposed in 1917 by Arthur Stanley Eddington[19] (who wrote at length on the dynamics of Cepheids), but it was not until 1953 that S. A. Zhevakin identified ionized helium as a likely valve for the engine.[20]

Classes

Cepheid variables are divided into two subclasses which exhibit markedly different masses, ages, and evolutionary histories: classical Cepheids and type II Cepheids. Delta Scuti variables are A-type stars on or near the main sequence at the lower end of the instability strip and were originally referred to as dwarf Cepheids. RR Lyrae variables have short periods and lie on the instability strip where it crosses the horizontal branch. Delta Scuti variables and RR Lyrae variables are not generally treated with Cepheid variables although their pulsations originate with the same helium ionisation kappa mechanism.

Classical Cepheids

Main page: Astronomy:Classical Cepheid variable
Light curve of Delta Cephei, the prototype of classical cepheids, showing the regular variations produced by intrinsic stellar pulsations

Classical Cepheids (also known as Population I Cepheids, type I Cepheids, or Delta Cepheid variables) undergo pulsations with very regular periods on the order of days to months. Classical Cepheids are Population I variable stars which are 4–20 times more massive than the Sun,[21] and up to 100,000 times more luminous.[22] These Cepheids are yellow bright giants and supergiants of spectral class F6 – K2 and their radii change by (~25% for the longer-period I Carinae) millions of kilometers during a pulsation cycle.[23]

Classical Cepheids are used to determine distances to galaxies within the Local Group and beyond, and are a means by which the Hubble constant can be established.[24][25][26][27][28] Classical Cepheids have also been used to clarify many characteristics of the Milky Way galaxy, such as the Sun's height above the galactic plane and the Galaxy's local spiral structure.[29]

A group of classical Cepheids with small amplitudes and sinusoidal light curves are often separated out as Small Amplitude Cepheids or s-Cepheids, many of them pulsating in the first overtone.

Type II Cepheids

Main page: Astronomy:Type II Cepheid
Light curve of κ Pavonis, a Type II cepheid, recorded by NASA's Transiting Exoplanet Survey Satellite (TESS)

Type II Cepheids (also termed Population II Cepheids) are population II variable stars which pulsate with periods typically between 1 and 50 days.[14][30] Type II Cepheids are typically metal-poor, old (~10 Gyr), low mass objects (~half the mass of the Sun). Type II Cepheids are divided into several subgroups by period. Stars with periods between 1 and 4 days are of the BL Her subclass, 10–20 days belong to the W Virginis subclass, and stars with periods greater than 20 days belong to the RV Tauri subclass.[14][30]

Type II Cepheids are used to establish the distance to the Galactic Center, globular clusters, and galaxies.[29][31][32][33][34][35][36]

Anomalous Cepheids

A group of pulsating stars on the instability strip have periods of less than 2 days, similar to RR Lyrae variables but with higher luminosities. Anomalous Cepheid variables have masses higher than type II Cepheids, RR Lyrae variables, and the Sun. It is unclear whether they are young stars on a "turned-back" horizontal branch, blue stragglers formed through mass transfer in binary systems, or a mix of both.[37][38]

Double-mode Cepheids

A small proportion of Cepheid variables have been observed to pulsate in two modes at the same time, usually the fundamental and first overtone, occasionally the second overtone.[39] A very small number pulsate in three modes, or an unusual combination of modes including higher overtones.[40]

Uncertain distances

Chief among the uncertainties tied to the classical and type II Cepheid distance scale are: the nature of the period-luminosity relation in various passbands, the impact of metallicity on both the zero-point and slope of those relations, and the effects of photometric contamination (blending with other stars) and a changing (typically unknown) extinction law on Cepheid distances. All these topics are actively debated in the literature.[25][22][27][34][41][42][43][44][45][46][47][48]

These unresolved matters have resulted in cited values for the Hubble constant (established from Classical Cepheids) ranging between 60 km/s/Mpc and 80 km/s/Mpc.[24][25][26][27][28] Resolving this discrepancy is one of the foremost problems in astronomy since the cosmological parameters of the Universe may be constrained by supplying a precise value of the Hubble constant.[26][28] Uncertainties have diminished over the years, due in part to discoveries such as RS Puppis.

Delta Cephei is also of particular importance as a calibrator of the Cepheid period-luminosity relation since its distance is among the most precisely established for a Cepheid, partly because it is a member of a star cluster[49][50] and the availability of precise Hubble Space Telescope/Hipparcos parallaxes.[51] The accuracy of parallax distance measurements to Cepheid variables and other bodies within 7,500 light-years is vastly improved by comparing images from Hubble taken six months apart when the Earth and Hubble are on opposite sides of the Sun.[52]

Pulsation model

Time lapse of the Cepheid type variable star Polaris illustrating the visual appearance of its cycle of brightness changes.

The accepted explanation for the pulsation of Cepheids is called the Eddington valve,[53][54] or "κ-mechanism", where the Greek letter κ (kappa) is the usual symbol for the gas opacity.

Helium is the gas thought to be most active in the process. Doubly ionized helium (helium whose atoms are missing both electrons) is more opaque than singly ionized helium. As helium is heated, its temperature rises until it reaches the point at which double ionisation spontaneously occurs and is sustained throughout the layer in much the same way a fluorescent tube 'strikes'. At the dimmest part of a Cepheid's cycle, this ionized gas in the outer layers of the star is relatively opaque, and so is heated by the star's radiation, and due to the increasing temperature, begins to expand. As it expands, it cools, but remains ionised until another threshold is reached at which point double ionization cannot be sustained and the layer becomes singly ionized hence more transparent, which allows radiation to escape. The expansion then stops, and reverses due to the star's gravitational attraction. The star's states are held to be either expanding or contracting by the hysterisis[55] generated by the doubly ionized helium and indefinitely flip-flops between the two states reversing every time the upper or lower threshold is crossed. This process is rather analogous to the relaxation oscillator found in electronics.[56]

In 1879, August Ritter (1826–1908) demonstrated that the adiabatic radial pulsation period for a homogeneous sphere is related to its surface gravity and radius through the relation:

[math]\displaystyle{ T = k \,\sqrt \frac R g }[/math]

where k is a proportionality constant. Now, since the surface gravity is related to the sphere mass and radius through the relation:

[math]\displaystyle{ g = k' \frac M {R^2} = k' \frac {RM} {R^3} = k' R\rho }[/math]

one finally obtains:

[math]\displaystyle{ T \sqrt \rho = Q }[/math]

where Q is a constant, called the pulsation constant.[57]

Examples

References

  1. ↑ Pigott, Edward (1785). "Observations of a new variable star". Philosophical Transactions of the Royal Society 75: 127–136. doi:10.1098/rstl.1785.0007. Bibcode: 1785RSPT...75..127P. 
  2. ↑ Goodricke, John (1786). "A series of observations on, and a discovery of, the period of the variation of the light of the star marked δ by Bayer, near the head of Cepheus. In a letter from John Goodricke, Esq. to Nevil Maskelyne, D.D.F.R.S. and Astronomer Royal". Philosophical Transactions of the Royal Society of London 76: 48–61. doi:10.1098/rstl.1786.0002. Bibcode: 1786RSPT...76...48G. https://books.google.com/books?id=0YMFAAAAQAAJ&pg=PA48. 
  3. ↑ Clarke, Agnes Mary (1903). Problems in Astrophysics. London, England: Adam & Charles Black. p. 319. ISBN 978-0-403-01478-1. https://books.google.com/books?id=8pQAAAAAMAAJ&pg=PA319. 
  4. ↑ Engle, Scott (2015). The Secret Lives of Cepheids: A Multi-Wavelength Study of the Atmospheres and Real-Time Evolution of Classical Cepheids (Thesis). arXiv:1504.02713. Bibcode:2015PhDT........45E. doi:10.5281/zenodo.45252.
  5. ↑ Leavitt, Henrietta S. (1908). "1777 variables in the Magellanic Clouds". Annals of the Astronomical Observatory of Harvard College 60 (4): 87–108. Bibcode: 1908AnHar..60...87L. https://babel.hathitrust.org/cgi/pt?id=coo.31924056092228&view=1up&seq=103. 
  6. ↑ Leavitt, Henrietta S.; Pickering, Edward C. (1912). "Periods of 25 variable stars in the Small Magellanic Cloud". Harvard College Observatory Circular 173: 1–3. Bibcode: 1912HarCi.173....1L. 
  7. ↑ Shapley, Harlow (December 1914). "On the Nature and Cause of Cepheid Variation". Astrophysical Journal 40: 448. doi:10.1086/142137. Bibcode: 1914ApJ....40..448S. 
  8. ↑ Shapley, H. (1916), "The variations in spectral type of twenty Cepheid variables", Astrophysical Journal 44: 273, doi:10.1086/142295, Bibcode: 1916ApJ....44..273S. 
  9. ↑ Hertzsprung, E. (1913). "Über die räumliche Verteilung der Veränderlichen vom δ Cephei-Typus" (in de). Astronomische Nachrichten 196 (4692): 201–208. Bibcode: 1913AN....196..201H. 
  10. ↑ Shapley, H. (1918). "Globular Clusters and the Structure of the Galactic System". Publications of the Astronomical Society of the Pacific 30 (173): 42. doi:10.1086/122686. Bibcode: 1918PASP...30...42S. 
  11. ↑ Hubble, E. P. (1925). "Cepheids in spiral nebulae". The Observatory 48: 139. Bibcode: 1925Obs....48..139H. 
  12. ↑ Lemaître, G. (1927). "Un Univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques". Annales de la Société Scientifique de Bruxelles 47: 49. Bibcode: 1927ASSB...47...49L. 
  13. ↑ "VISTA Discovers New Component of Milky Way". http://www.eso.org/public/news/eso1542/. 
  14. ↑ 14.0 14.1 14.2 Wallerstein, George (2002). "The Cepheids of Population II and Related Stars". Publications of the Astronomical Society of the Pacific 114 (797): 689–699. doi:10.1086/341698. Bibcode: 2002PASP..114..689W. 
  15. ↑ Baade, W. (1958). "Problems in the determination of the distance of galaxies". Astronomical Journal 63: 207. doi:10.1086/107726. Bibcode: 1958AJ.....63..207B. 
  16. ↑ Allen, Nick. "Section 2: The Great Debate and the Great Mistake: Shapley, Hubble, Baade". The Cepheid Distance Scale: A History. http://www.institute-of-brilliant-failures.com/section2.htm. 
  17. ↑ Shapley, Harlow. (1918). "No. 153. Studies based on the colors and magnitudes in stellar clusters. Eighth paper: The luminosities and distances of 139 Cepheid variables". Contributions from the Mount Wilson Observatory 153: 1. Bibcode: 1918CMWCI.153....1S. 
  18. ↑ Shapley, Harlow (1918). "Studies based on the colors and magnitudes in stellar clusters. Eighth paper: The luminosities and distances of 139 Cepheid variables". Astrophysical Journal 48: 279–294. doi:10.1086/142435. Bibcode: 1918ApJ....48..279S. 
  19. ↑ Eddington, A. S. (1917). "The pulsation theory of Cepheid variables". The Observatory 40: 290. Bibcode: 1917Obs....40..290E. 
  20. ↑ Zhevakin, S. A. (1953). "К Теории Цефеид. I". Астрономический журнал 30: 161–179. 
  21. ↑ Turner, David G. (1996). "The Progenitors of Classical Cepheid Variables". Journal of the Royal Astronomical Society of Canada 90: 82. Bibcode: 1996JRASC..90...82T. 
  22. ↑ 22.0 22.1 Turner, David G. (2010). "The PL calibration for Milky Way Cepheids and its implications for the distance scale". Astrophysics and Space Science 326 (2): 219–231. doi:10.1007/s10509-009-0258-5. Bibcode: 2010Ap&SS.326..219T. 
  23. ↑ Rodgers, A. W. (1957). "Radius variation and population type of cepheid variables". Monthly Notices of the Royal Astronomical Society 117: 85–94. doi:10.1093/mnras/117.1.85. Bibcode: 1957MNRAS.117...85R. 
  24. ↑ 24.0 24.1 Freedman, Wendy L.; Madore, Barry F.; Gibson, Brad K.; Ferrarese, Laura; Kelson, Daniel D.; Sakai, Shoko; Mould, Jeremy R.; Kennicutt, Jr., Robert C. et al. (2001). "Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant". The Astrophysical Journal 553 (1): 47–72. doi:10.1086/320638. Bibcode: 2001ApJ...553...47F. 
  25. ↑ 25.0 25.1 25.2 Tammann, G. A.; Sandage, A.; Reindl, B. (2008). "The expansion field: the value of H 0". The Astronomy and Astrophysics Review 15 (4): 289–331. doi:10.1007/s00159-008-0012-y. Bibcode: 2008A&ARv..15..289T. 
  26. ↑ 26.0 26.1 26.2 Freedman, Wendy L.; Madore, Barry F. (2010). "The Hubble Constant". Annual Review of Astronomy and Astrophysics 48: 673–710. doi:10.1146/annurev-astro-082708-101829. Bibcode: 2010ARA&A..48..673F. 
  27. ↑ 27.0 27.1 27.2 Ngeow, C.; Kanbur, S. M. (2006). "The Hubble Constant from Type Ia Supernovae Calibrated with the Linear and Nonlinear Cepheid Period-Luminosity Relations". The Astrophysical Journal 642 (1): L29–L32. doi:10.1086/504478. Bibcode: 2006ApJ...642L..29N. 
  28. ↑ 28.0 28.1 28.2 Macri, Lucas M.; Riess, Adam G.; Guzik, Joyce Ann; Bradley, Paul A. (2009). "AIP Conference Proceedings". Stellar Pulsation: Challenges for Theory and Observation: Proceedings of the International Conference. AIP Conference Proceedings. 1170. pp. 23–25. doi:10.1063/1.3246452. Bibcode: 2009AIPC.1170...23M. 
  29. ↑ 29.0 29.1 Majaess, D. J.; Turner, D. G.; Lane, D. J. (2009). "Characteristics of the Galaxy according to Cepheids". Monthly Notices of the Royal Astronomical Society 398 (1): 263–270. doi:10.1111/j.1365-2966.2009.15096.x. Bibcode: 2009MNRAS.398..263M. 
  30. ↑ 30.0 30.1 Soszyński, I.; Udalski, A.; Szymański, M. K.; Kubiak, M.; Pietrzyński, G.; Wyrzykowski, Ł.; Szewczyk, O.; Ulaczyk, K. et al. (2008). "The Optical Gravitational Lensing Experiment. The OGLE-III Catalog of Variable Stars. II.Type II Cepheids and Anomalous Cepheids in the Large Magellanic Cloud". Acta Astronomica 58: 293. Bibcode: 2008AcA....58..293S. 
  31. ↑ Kubiak, M.; Udalski, A. (2003). "The Optical Gravitational Lensing Experiment. Population II Cepheids in the Galactic Bulge". Acta Astronomica 53: 117. Bibcode: 2003AcA....53..117K. 
  32. ↑ Matsunaga, Noriyuki; Fukushi, Hinako; Nakada, Yoshikazu; Tanabé, Toshihiko; Feast, Michael W.; Menzies, John W.; Ita, Yoshifusa; Nishiyama, Shogo et al. (2006). "The period-luminosity relation for type II Cepheids in globular clusters". Monthly Notices of the Royal Astronomical Society 370 (4): 1979–1990. doi:10.1111/j.1365-2966.2006.10620.x. Bibcode: 2006MNRAS.370.1979M. 
  33. ↑ Feast, Michael W.; Laney, Clifton D.; Kinman, Thomas D.; Van Leeuwen, Floor; Whitelock, Patricia A. (2008). "The luminosities and distance scales of type II Cepheid and RR Lyrae variables". Monthly Notices of the Royal Astronomical Society 386 (4): 2115–2134. doi:10.1111/j.1365-2966.2008.13181.x. Bibcode: 2008MNRAS.386.2115F. 
  34. ↑ 34.0 34.1 Majaess, D.; Turner, D.; Lane, D. (2009). "Type II Cepheids as Extragalactic Distance Candles". Acta Astronomica 59 (4): 403. Bibcode: 2009AcA....59..403M. 
  35. ↑ Majaess, D. J. (2010). "RR Lyrae and Type II Cepheid Variables Adhere to a Common Distance Relation". The Journal of the American Association of Variable Star Observers 38 (1): 100–112. Bibcode: 2010JAVSO..38..100M. 
  36. ↑ Matsunaga, Noriyuki; Feast, Michael W.; Menzies, John W. (2009). "Period-luminosity relations for type II Cepheids and their application". Monthly Notices of the Royal Astronomical Society 397 (2): 933–942. doi:10.1111/j.1365-2966.2009.14992.x. Bibcode: 2009MNRAS.397..933M. 
  37. ↑ Caputo, F.; Castellani, V.; Degl'Innocenti, S.; Fiorentino, G.; Marconi, M. (2004). "Bright metal-poor variables: Why Anomalous Cepheids?". Astronomy and Astrophysics 424 (3): 927–934. doi:10.1051/0004-6361:20040307. Bibcode: 2004A&A...424..927C. 
  38. ↑ Soszyński, I.; Udalski, A.; Szymański, M. K.; Kubiak, M.; Pietrzyński, G.; Wyrzykowski, Ł.; Szewczyk, O.; Ulaczyk, K. et al. (2008). "The Optical Gravitational Lensing Experiment. The OGLE-III Catalog of Variable Stars. II.Type II Cepheids and Anomalous Cepheids in the Large Magellanic Cloud". Acta Astronomica 58: 293. Bibcode: 2008AcA....58..293S. 
  39. ↑ Smolec, R.; Moskalik, P. (2008). "Double-Mode Classical Cepheid Models, Revisited". Acta Astronomica 58: 233. Bibcode: 2008AcA....58..233S. 
  40. ↑ Soszynski, I.; Poleski, R.; Udalski, A.; Kubiak, M.; Szymanski, M. K.; Pietrzynski, G.; Wyrzykowski, L.; Szewczyk, O. et al. (2008). "The Optical Gravitational Lensing Experiment. Triple-Mode and 1O/3O Double-Mode Cepheids in the Large Magellanic Cloud". Acta Astronomica 58: 153. Bibcode: 2008AcA....58..153S. 
  41. ↑ Benedict, G. Fritz; McArthur, Barbara E.; Feast, Michael W.; Barnes, Thomas G.; Harrison, Thomas E.; Patterson, Richard J.; Menzies, John W.; Bean, Jacob L. et al. (2007). "Hubble Space Telescope Fine Guidance Sensor Parallaxes of Galactic Cepheid Variable Stars: Period-Luminosity Relations". The Astronomical Journal 133 (4): 1810. doi:10.1086/511980. Bibcode: 2007AJ....133.1810B. 
  42. ↑ Stanek, K. Z.; Udalski, A. (1999). "The Optical Gravitational Lensing Experiment. Investigating the Influence of Blending on the Cepheid Distance Scale with Cepheids in the Large Magellanic Cloud". arXiv:astro-ph/9909346.
  43. ↑ Udalski, A.; Wyrzykowski, L.; Pietrzynski, G.; Szewczyk, O.; Szymanski, M.; Kubiak, M.; Soszynski, I.; Zebrun, K. (2001). "The Optical Gravitational Lensing Experiment. Cepheids in the Galaxy IC1613: No Dependence of the Period-Luminosity Relation on Metallicity". Acta Astronomica 51: 221. Bibcode: 2001AcA....51..221U. 
  44. ↑ Macri, L. M.; Stanek, K. Z.; Bersier, D.; Greenhill, L. J.; Reid, M. J. (2006). "A New Cepheid Distance to the Maser‐Host Galaxy NGC 4258 and Its Implications for the Hubble Constant". The Astrophysical Journal 652 (2): 1133–1149. doi:10.1086/508530. Bibcode: 2006ApJ...652.1133M. 
  45. ↑ Bono, G.; Caputo, F.; Fiorentino, G.; Marconi, M.; Musella, I. (2008). "Cepheids in External Galaxies. I. The Maser‐Host Galaxy NGC 4258 and the Metallicity Dependence of Period‐Luminosity and Period‐Wesenheit Relations". The Astrophysical Journal 684 (1): 102–117. doi:10.1086/589965. Bibcode: 2008ApJ...684..102B. 
  46. ↑ Madore, Barry F.; Freedman, Wendy L. (2009). "Concerning the Slope of the Cepheid Period-Luminosity Relation". The Astrophysical Journal 696 (2): 1498–1501. doi:10.1088/0004-637X/696/2/1498. Bibcode: 2009ApJ...696.1498M. 
  47. ↑ Scowcroft, V.; Bersier, D.; Mould, J. R.; Wood, P. R. (2009). "The effect of metallicity on Cepheid magnitudes and the distance to M33". Monthly Notices of the Royal Astronomical Society 396 (3): 1287–1296. doi:10.1111/j.1365-2966.2009.14822.x. Bibcode: 2009MNRAS.396.1287S. 
  48. ↑ Majaess, D. (2010). "The Cepheids of Centaurus A (NGC 5128) and Implications for H0". Acta Astronomica 60 (2): 121. Bibcode: 2010AcA....60..121M. 
  49. ↑ De Zeeuw, P. T.; Hoogerwerf, R.; De Bruijne, J. H. J.; Brown, A. G. A.; Blaauw, A. (1999). "A HIPPARCOS Census of the Nearby OB Associations". The Astronomical Journal 117 (1): 354–399. doi:10.1086/300682. Bibcode: 1999AJ....117..354D. 
  50. ↑ Majaess, D.; Turner, D.; Gieren, W. (2012). "New Evidence Supporting Cluster Membership for the Keystone Calibrator Delta Cephei". The Astrophysical Journal 747 (2): 145. doi:10.1088/0004-637X/747/2/145. Bibcode: 2012ApJ...747..145M. 
  51. ↑ Benedict, G. Fritz; McArthur, B. E.; Fredrick, L. W.; Harrison, T. E.; Slesnick, C. L.; Rhee, J.; Patterson, R. J.; Skrutskie, M. F. et al. (2002). "Astrometry with the Hubble Space Telescope: A Parallax of the Fundamental Distance Calibrator δ Cephei". The Astronomical Journal 124 (3): 1695. doi:10.1086/342014. Bibcode: 2002AJ....124.1695B. 
  52. ↑ Riess, Adam G.; Casertano, Stefano; Anderson, Jay; MacKenty, John; Filippenko, Alexei V. (2014). "Parallax beyond a Kiloparsec from Spatially Scanning the Wide Field Camera 3 on the Hubble Space Telescope". The Astrophysical Journal 785 (2): 161. doi:10.1088/0004-637X/785/2/161. Bibcode: 2014ApJ...785..161R. 
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  54. ↑ "American Association of Variable Star Observers". The Encyclopedia of Astronomy and Astrophysics. 2001. doi:10.1888/0333750888/4130. ISBN 0-333-75088-8. 
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  56. ↑ "Relaxation oscillator" (in en), Wikipedia, 2023-05-10, https://en.wikipedia.org/w/index.php?title=Relaxation_oscillator&oldid=1154083763, retrieved 2023-09-04 
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  60. ↑ Plachy, E. et al. (2020), "TESS observations of Cepheid stars: first light results", The Astrophysical Journal Supplement Series 253 (1): 11, doi:10.3847/1538-4365/abd4e3, Bibcode: 2021ApJS..253...11P 

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